A minimization of a curvature functional on fiber bundles

Let B be a smooth compact orientable surface without boundary and with $\chi(B) < 0.$ We examine two types of fiber bundles M over B with fiber F. The first is a principle fiber bundle with a two-torus fiber and the second is an $S\sp2$ fiber bundle with an SO(3) group action. In each case, the tangent space of the bundle can be decomposed into a vertical space, those vectors tangent to fibers, and a horizontal space complementary to the vertical space and invariant under the group action. The bundle can be given a metric that is the direct sum of metrics on the vertical and horizontal spaces. Additionally, with this metric, M, is locally isometric to a product space $B\times F$ with metric $g\sb{b} + g\sb{f}.$ Here $g\sb{b}$ is any fixed metric on the base, $g\sb{f}$ is a constant curvature metric on the fiber invariant under the action of the group. We can obtain a new metric on M by scaling the horizontal component of the original by $e\sp{2u}$ and the vertical component by $f\sp2,$ where u and f are smooth functions on the base. We put certain constraints on u and f and consider the family of all such variations. In this thesis, we show, using nonlinear elliptic estimates, that among these metrics there is one for which the integral of the norm of the Ricci curvature tensor squared, $\int\sb{M}\vert Ric\vert\sp2dV,$ is minimized.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Natural Sciences

Grantor

Rice University

Year dc:date.issued

1998

Author dc:creator

• Hawkins, Christopher Ryan

Advisor dc:contributor.advisor

• Gao, Zhiyong

• Mathematics

Statement dc:rights

• Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1911/19268

OAI identifier oai:identifier

oai:repository.rice.edu:1911/19268